We present an exact and efficient algorithm for computing the Reeb space of a bivariate PL map. The Reeb space is a topological structure that generalizes the Reeb graph to the setting of multiple scalar-valued functions defined over a shared domain, a situation that frequently arises in practical applications. While the Reeb graph has become a standard tool in computer graphics, shape analysis, and scientific visualization, the Reeb space is still in the early stages of adoption. Although several algorithms for computing the Reeb space have been proposed, none offer an implementation that is both exact and efficient, which has substantially limited its practical use. To address this gap, we introduce singular arrange and traverse, a new algorithm built upon the arrange and traverse framework [Hristov et al., 2025]. Our method exploits the fact that, in the bivariate case, only singular edges contribute to the structure of Reeb space, allowing us to ignore many regular edges [Tierny and Carr, 2017]. This observation results in substantial efficiency gains on datasets where most edges are regular, which is common in many numerical simulations of physical systems. We provide an implementation of our method and benchmark it against the original arrange and traverse algorithm, showing performance gains of up to four orders of magnitude on real-world datasets.

Abstract Uncertainty in experimental measurements makes it challenging to determine which features are intrinsic to the phenomenon and which are most stable and reliable. Granular materials, such as sand, form a complex system in which the forces between individual particles influence the material's macroscopic behavior. However, these forces are also subject to uncertainty, as repeated measurements can yield different results. In this paper, we investigate how to model and visually analyze the uncertain structure of forces in granular materials. We adopt different perspectives on uncertainty by considering it as variance, probability, or additional variable. For a nuanced analysis of granular material data, we propose combining visualizations that represent these perspectives into a single visual analytics approach. We integrate uncertainty‐aware spatial visualizations that convey the probability of features, visualizations of derived measures and their variance over changes in the packing fraction, and overviews of varying probability thresholds. Finally, we evaluate our proposed approach in a case study conducted together with geotechnical engineers for the example of a 2D ensemble of photoelastic disks.

This paper presents an interactive analysis framework for exploring data from photoelastic disk experiments, which serve as a model for two-dimensional granular materials. Granular materials, composed of discrete particles such as sand or gravel, exhibit behaviors resembling fluid or solid states depending on the system configuration. These behaviors arise from interparticle contact forces, which form complex force networks that govern the material's macroscopic behavior. Our framework is specifically designed to analyze such 2D ensembles of dynamic force networks, enabling the identification and characterization of their underlying structures. The framework is built around a topology-based, multiscale data segmentation in terms of force chains and cycles. The analysis methods are structured across three levels: (1) multiscale analysis of individual instances under specific loading conditions, (2) detailed exploration of single experiments encompassing a series of loading and unloading cycles, and (3) comparative analysis across experiments conducted under similar and differing setups. We demonstrate the capabilities of our framework with a case study for each of these levels.

Abstract We present the first combinatorial algorithm for efficiently computing the Reeb space in all dimensions. The Reeb space is a higher‐dimensional generalization of the Reeb graph, which is standard practice in the analysis of scalar fields, along with other computational topology tools such as persistent homology and the Morse‐Smale complex. One significant limitation of topological tools for scalar fields is that data often involves multiple variables, where joint analysis is more insightful. Generalizing topological data structures to multivariate data has proven challenging and the Reeb space is one of the few available options. However, none of the existing algorithms can efficiently compute the Reeb space in arbitrary dimensions and there are no available implementations which are robust with respect to numerical errors. We propose a new algorithm for computing the Reeb space of a generic piecewise linear map over a simplicial mesh of any dimension called arrange and traverse. We implement a robust specialization of our algorithm for tetrahedral meshes and evaluate it on real‐life data.

Abstract Oceanographers rely on visual analysis to interpret model simulations, identify events and phenomena, and track dynamic ocean processes. The ever increasing resolution and complexity of ocean data due to its dynamic nature and multivariate relationships demands a scalable and adaptable visualization tool for interactive exploration. We introduce pyParaOcean, a scalable and interactive visualization system designed specifically for ocean data analysis. pyParaOcean offers specialized modules for common oceanographic analysis tasks, including eddy identification and salinity movement tracking. These modules seamlessly integrate with ParaView as filters, ensuring a user‐friendly and easy‐to‐use system while leveraging the parallelization capabilities of ParaView and a plethora of inbuilt general‐purpose visualization functionalities. The creation of an auxiliary dataset stored as a Cinema database helps address I/O and network bandwidth bottlenecks while supporting the generation of quick overview visualizations. We present a case study on the Bay of Bengal to demonstrate the utility of the system and scaling studies to evaluate the efficiency of the system.

Feature level sets (FLS) have shown significant potential in the analysis of multi-field data by using traits defined in attribute space to specify features in the domain. In this work, we address key challenges in the practical use of FLS: trait design and feature selection for rendering. To simplify trait design, we propose a Cartesian decomposition of traits into simpler components, making the process more intuitive and computationally efficient. Additionally, we utilize dictionary learning results to automatically suggest point traits. To enhance feature selection, we introduce trait-induced merge trees (TIMTs), a generalization of merge trees for feature level sets, aimed at topologically analyzing tensor fields or general multi-variate data. The leaves in the TIMT represent areas in the input data that are closest to the defined trait, thereby most closely resembling the defined feature. This merge tree provides a hierarchy of features, enabling the querying of the most relevant and persistent features. Our method includes various query techniques for the tree, allowing the highlighting of different aspects. We demonstrate the cross-application capabilities of this approach through five case studies from different domains.

Photoinduced electronic transitions are complex quantum-mechanical processes where electrons move between energy levels due to the absorption of light. This induces dynamics i.e., coupled changes in the electronic structure and nuclear geometry, that drive physical and chemical processes of importance in diverse fields ranging from photobiology and materials design to medicine. The evolving electronic structure can be characterized by two electron density fields: hole and particle natural transition orbitals (NTOs). A study of the two density fields helps understand the movement of electronic charge from one part of the molecule to another, specifically the donor and acceptor regions. Previous works in this area rely on side-by-side visual comparisons of isosurfaces, statistical approaches, or visual analysis of bivariate fields restricted to limited time instances. We propose a new method to analyze time-varying bivariate fields with a large number of instances, as pertinent to understand electronic structure changes during light-induced dynamics. Since the NTO fields depend on the nuclear geometry, the nuclear motion leads to a large number of bivariate field instances. Structures like tracking graphs have been used to analyze time-varying univariate fields. This article presents a structured and practical approach to feature-directed visual exploration of time-varying bivariate fields using continuous scatterplots (CSPs) and image moment-based descriptors, tailored for studying the evolving electronic structure following photoexcitation. The CSP of the bivariate field at every time step is represented using an image moment vector of length 4. The collection of all image moment vector descriptors is considered as a point cloud in $\mathbb {R}^{4}$R4 and visualized using principal component analysis. Choosing an appropriate pair of principal components results in a representation of the point cloud as a curve on the plane. This representation supports tasks such as identifying interesting time steps, identifying patterns within the bivariate field, and tracking their evolution over time. We present two case studies on excited-state dynamics in molecular systems that demonstrate how the time-varying bivariate field analysis helps provide application-specific insights.

Abstract Diffusion‐weighted magnetic resonance imaging (D‐MRI) is a technique to measure the diffusion of water, in biological tissues. It is used to detect microscopic patterns, such as neural fibers in the living human brain, with many medical and neuroscience applications e.g. for fiber tracking. In this paper, we consider High‐Angular Resolution Diffusion Imaging (HARDI) which provides one of the richest representations of water diffusion. It records the movement of water molecules by measuring diffusion under 64 or more directions. A key challenge is that it generates high‐dimensional, large, and complex datasets. In our work, we develop a novel representation that exploits the inherent sparsity of the HARDI signal by approximating it as a linear sum of basic atoms in an overcomplete data‐driven dictionary using only a sparse set of coefficients. We show that this approach can be efficiently integrated into the standard q‐ball imaging pipeline to compute the diffusion orientation distribution function (ODF). Sparse representations have the potential to reduce the size of the data while also giving some insight into the data. To explore the results, we provide a visualization of the atoms of the dictionary and their frequency in the data to highlight the basic characteristics of the data. We present our proposed pipeline and demonstrate its performance on 5 HARDI datasets.

Electronic transitions in molecules due to the absorption or emission of light is a complex quantum mechanical process. Their study plays an important role in the design of novel materials. A common yet challenging task in the study is to determine the nature of electronic transitions, namely which subgroups of the molecule are involved in the transition by donating or accepting electrons, followed by an investigation of the variation in the donor-acceptor behavior for different transitions or conformations of the molecules. In this article, we present a novel approach for the analysis of a bivariate field and show its applicability to the study of electronic transitions. This approach is based on two novel operators, the continuous scatterplot (CSP) lens operator and the CSP peel operator, that enable effective visual analysis of bivariate fields. Both operators can be applied independently or together to facilitate analysis. The operators motivate the design of control polygon inputs to extract fiber surfaces of interest in the spatial domain. The CSPs are annotated with a quantitative measure to further support the visual analysis. We study different molecular systems and demonstrate how the CSP peel and CSP lens operators help identify and study donor and acceptor characteristics in molecular systems.

Merge trees, a type of topological descriptors, serve to identify and summarize the topological characteristics associated with scalar fields. They have great potential for analyzing and visualizing time-varying data. First, they give compressed and topology-preserving representations of data instances. Second, their comparisons provide a basis for studying the relations among data instances, such as their distributions, clusters, outliers, and periodicities. A number of comparative measures have been developed for merge trees. However, these measures are often computationally expensive since they implicitly consider all possible correspondences between critical points of the merge trees. In this paper, we perform geometry-aware comparisons of merge trees using labeled interleaving distances. The main idea is to decouple the computation of a comparative measure into two steps: a labeling step that generates a correspondence between the critical points of two merge trees, and a comparison step that computes distances between a pair of labeled merge trees by encoding them as matrices. We show that our approach is general, computationally efficient, and practically useful. Our framework makes it possible to integrate geometric information of the data domain in the labeling process. At the same time, the framework reduces the computational complexity since not all possible correspondences have to be considered. We demonstrate via experiments that such geometry-aware merge tree comparisons help to detect transitions, clusters, and periodicities of time-varying datasets, as well as to diagnose and highlight the topological changes between adjacent data instances.

We present a visual analysis environment based on a multi-scale partitioning of a 2d domain into regions bounded by cycles in weighted planar embedded graphs. The work has been inspired by an application in granular materials research, where the question of scale plays a fundamental role in the analysis of material properties. We propose an efficient algorithm to extract the hierarchical cycle structure using persistent homology. The core of the algorithm is a filtration on a dual graph exploiting Alexander’s duality. The resulting partitioning is the basis for the derivation of statistical properties that can be explored in a visual environment. We demonstrate the proposed pipeline on a few synthetic and one real-world data set.

Abstract We present a pipeline for the interactive visual analysis and exploration of molecular electronic transition ensembles. Each ensemble member is specified by a molecular configuration, the charge transfer between two molecular states, and a set of physical properties. The pipeline is targeted towards theoretical chemists, supporting them in comparing and characterizing electronic transitions by combining automatic and interactive visual analysis. A quantitative feature vector characterizing the electron charge transfer serves as the basis for hierarchical clustering as well as for the visual representations. The interface for the visual exploration consists of four components. A dendrogram provides an overview of the ensemble. It is augmented with a level of detail glyph for each cluster. A scatterplot using dimensionality reduction provides a second visualization, highlighting ensemble outliers. Parallel coordinates show the correlation with physical parameters. A spatial representation of selected ensemble members supports an in‐depth inspection of transitions in a form that is familiar to chemists. All views are linked and can be used to filter and select ensemble members. The usefulness of the pipeline is shown in three different case studies.

In molecular analysis, spatial distribution functions (SDF) are fundamental instruments in answering questions related to spatial occurrences and relations of atomic structures over time. Given a molecular trajectory, SDFs can, for example, reveal the occurrence of water in relation to particular structures and hence provide clues of hydrophobic and hydrophilic regions. For the computation of meaningful distribution functions, the definition of molecular reference structures is essential. Therefore we introduce the concept of an internal frame of reference (IFR) for labeled point sets that represent selected molecular structures, and we propose an algorithm for tracking the IFR over time and space using a variant of Kabsch's algorithm. This approach lets us generate a consistent space for the aggregation of the SDF for molecular trajectories and molecular ensembles. We demonstrate the usefulness of the technique by applying it to temporal molecular trajectories as well as ensemble datasets. The examples include different docking scenarios with DNA, insulin, and aspirin.

Abstract We propose a novel method for the computation of Jacobi sets in 2D domains. The Jacobi set is a topological descriptor based on Morse theory that captures gradient alignments among multiple scalar fields, which is useful for multi-field visualization. Previous Jacobi set computations use piecewise linear approximations on triangulations that result in discretization artifacts like zig-zag patterns. In this paper, we utilize a local bilinear method to obtain a more precise approximation of Jacobi sets by preserving the topology and improving the geometry. Consequently, zig-zag patterns on edges are avoided, resulting in a smoother Jacobi set representation. Our experiments show a better convergence with increasing resolution compared to the piecewise linear method. We utilize this advantage with an efficient local subdivision scheme. Finally, our approach is evaluated qualitatively and quantitatively in comparison with previous methods for different mesh resolutions and across a number of synthetic and real-world examples.

Topological and geometric segmentation methods provide powerful concepts for detailed field analysis and visualization. However, when it comes to a quantitative analysis that requires highly accurate geometric segmentation, there is a large discrepancy between the promising theory and the available computational approaches. In this paper, we compare and evaluate various segmentation methods with the aim to identify and quantify the extent of these discrepancies. Thereby, we focus on an application from quantum chemistry: the analysis of electron density fields. It is a scalar quantity that can be experimentally measured or theoretically computed. In the evaluation we consider methods originating from the domain of quantum chemistry and computational topology. We apply the methods to the charge density of a set of crystals and molecules. Therefore, we segment the volumes into atomic regions and derive and compare quantitative measures such as total charge and dipole moments from these regions. As a result, we conclude that an accurate geometry determination can be crucial for correctly segmenting and analyzing a scalar field, here demonstrated on the electron density field.

Abstract Tensors are used to describe complex physical processes in many applications. Examples include the distribution of stresses in technical materials, acting forces during seismic events, or remodeling of biological tissues. While tensors encode such complex information mathematically precisely, the semantic interpretation of a tensor is challenging. Visualization can be beneficial here and is frequently used by domain experts. Typical strategies include the use of glyphs, color plots, lines, and isosurfaces. However, data complexity is nowadays accompanied by the sheer amount of data produced by large‐scale simulations and adds another level of obstruction between user and data. Given the limitations of traditional methods, and the extra cognitive effort of simple methods, more advanced tensor field visualization approaches have been the focus of this work. This survey aims to provide an overview of recent research results with a strong application‐oriented focus, targeting applications based on continuum mechanics, namely the fields of structural, bio‐, and geomechanics. As such, the survey is complementing and extending previously published surveys. Its utility is twofold: (i) It serves as basis for the visualization community to get an overview of recent visualization techniques. (ii) It emphasizes and explains the necessity for further research for visualizations in this context.

Abstract The study of electronic transitions within a molecule connected to the absorption or emission of light is a common task in the process of the design of new materials. The transitions are complex quantum mechanical processes and a detailed analysis requires a breakdown of these processes into components that can be interpreted via characteristic chemical properties. We approach these tasks by providing a detailed analysis of the electron density field. This entails methods to quantify and visualize electron localization and transfer from molecular subgroups combining spatial and abstract representations. The core of our method uses geometric segmentation of the electronic density field coupled with a graph‐theoretic formulation of charge transfer between molecular subgroups. The design of the methods has been guided by the goal of providing a generic and objective analysis following fundamental concepts. We illustrate the proposed approach using several case studies involving the study of electronic transitions in different molecular systems.

Abstract In topological data analysis and visualization, topological descriptors such as persistence diagrams, merge trees, contour trees, Reeb graphs, and Morse–Smale complexes play an essential role in capturing the shape of scalar field data. We present a state‐of‐the‐art report on scalar field comparison using topological descriptors. We provide a taxonomy of existing approaches based on visualization tasks associated with three categories of data: single fields, time‐varying fields, and ensembles. These tasks include symmetry detection, periodicity detection, key event/feature detection, feature tracking, clustering, and structure statistics. Our main contributions include the formulation of a set of desirable mathematical and computational properties of comparative measures, and the classification of visualization tasks and applications that are enabled by these measures.

Abstract We present a state‐of‐the‐art report on time‐dependent flow topology. We survey representative papers in visualization and provide a taxonomy of existing approaches that generalize flow topology from time‐independent to time‐dependent settings. The approaches are classified based upon four categories: tracking of steady topology, reference frame adaption, pathline classification or clustering, and generalization of critical points. Our unique contributions include introducing a set of desirable mathematical properties to interpret physical meaningfulness for time‐dependent flow visualization, inferring mathematical properties associated with selective research papers, and utilizing such properties for classification. The five most important properties identified in the existing literature include coincidence with the steady case, induction of a partition within the domain, Lagrangian invariance, objectivity, and Galilean invariance.

Abstract We present an application, and its development process, for interactive visual analysis of brain imaging data and clinical measurements. The application targets neuroscientists interested in understanding the correlations between active brain regions and physiological or psychological factors. The application has been developed in a participatory design process and has subsequently been released as the free software ‘VisualNeuro’. From initial observations of the neuroscientists' workflow, we concluded that while existing tools provide powerful analysis options, they lack effective interactive exploration requiring the use of many tools side by side. Consequently, our application has been designed to simplify the workflow combining statistical analysis with interactive visual exploration. The resulting environment comprises parallel coordinates for effective overview and selection, Welch's t‐test to filter out brain regions with statistically significant differences and multiple visualizations for comparison between brain regions and clinical parameters. These exploration concepts enable neuroscientists to interactively explore the complex bidirectional interplay between clinical and brain measurements and easily compare different patient groups. A qualitative user study has been performed with three neuroscientists from different domains. The study shows that the developed environment supports simultaneous analysis of more parameters, provides rapid pathways to insights and is an effective tool for hypothesis formation.

Iso-surfaces or level-sets provide an effective and frequently used means for feature visualization. However, they are restricted to simple features for uni-variate data. The approach does not scale when moving to multi-variate data or when considering more complex feature definitions. In this paper, we introduce the concept of traits and feature level-sets, which can be understood as a generalization of level-sets as it includes iso-surfaces, and fiber surfaces as special cases. The concept is applicable to a large class of traits defined as subsets in attribute space, which can be arbitrary combinations of points, lines, surfaces and volumes. It is implemented into a system that provides an interface to define traits in an interactive way and multiple rendering options. We demonstrate the effectiveness of the approach using multi-variate data sets of different nature, including vector and tensor data, from different application domains.

The complexity of today's visualization applications demands specific visualization systems tailored for the development of these applications. Frequently, such systems utilize levels of abstraction to improve the application development process, for instance by providing a data flow network editor. Unfortunately, these abstractions result in several issues, which need to be circumvented through an abstraction-centered system design. Often, a high level of abstraction hides low level details, which makes it difficult to directly access the underlying computing platform, which would be important to achieve an optimal performance. Therefore, we propose a layer structure developed for modern and sustainable visualization systems allowing developers to interact with all contained abstraction levels. We refer to this interaction capabilities as usage abstraction levels, since we target application developers with various levels of experience. We formulate the requirements for such a system, derive the desired architecture, and present how the concepts have been exemplary realized within the Inviwo visualization system. Furthermore, we address several specific challenges that arise during the realization of such a layered architecture, such as communication between different computing platforms, performance centered encapsulation, as well as layer-independent development by supporting cross layer documentation and debugging capabilities.

Abstract We present an interactive approach to analyse flow fields using a new type of particle system, which is composed of autonomous particles exploring the flow. While particles provide a very intuitive way to visualize flows, it is a challenge to capture the important features with such systems. Particles tend to cluster in regions of low velocity and regions of interest are often sparsely populated. To overcome these disadvantages, we propose an automatic adaption of the particle density with respect to local importance measures. These measures are user defined and the systems sensitivity to them can be adjusted interactively. Together with the particle history, these measures define a probability for particles to multiply or die, respectively. There is no communication between the particles and no neighbourhood information has to be maintained. Thus, the particles can be handled in parallel and support a real‐time investigation of flow fields. To enhance the visualization, the particles' properties and selected field measures are also used to specify the systems rendering parameters, such as colour and size. We demonstrate the effectiveness of our approach on different simulated vector fields from technical and medical applications.

Abstract In this work, we propose a controlled simplification strategy for degenerated points in symmetric 2D tensor fields that is based on the topological notion of robustness . Robustness measures the structural stability of the degenerate points with respect to variation in the underlying field. We consider an entire pipeline for generating a hierarchical set of degenerate points based on their robustness values. Such a pipeline includes the following steps: the stable extraction and classification of degenerate points using an edge labeling algorithm, the computation and assignment of robustness values to the degenerate points, and the construction of a simplification hierarchy. We also discuss the challenges that arise from the discretization and interpolation of real world data.

Abstract In the field of organic electronics, understanding complex material morphologies and their role in efficient charge transport in solar cells is extremely important. Related processes are studied using the Ising model and Kinetic Monte Carlo simulations resulting in large ensembles of stochastic trajectories. Naive visualization of these trajectories, individually or as a whole, does not lead to new knowledge discovery through exploration. In this paper, we present novel visualization and exploration methods to analyze this complex dynamic data, which provide succinct and meaningful abstractions leading to scientific insights. We propose a morphology abstraction yielding a network composed of material pockets and the interfaces, which serves as backbone for the visualization of the charge diffusion. The trajectory network is created using a novel way of implicitly attracting the trajectories to the skeleton of the morphology relying on a relaxation process. Each individual trajectory is then represented as a connected sequence of nodes in the skeleton. The final network summarizes all of these sequences in a single aggregated network. We apply our method to three different morphologies and demonstrate its suitability for exploring this kind of data.

Analyzing depressions plays an important role in meteorology, especially in the study of cyclones. In particular, the study of the temporal evolution of cyclones requires a robust depression tracking framework. To cope with this demand we propose a pipeline for the exploration of cyclones and their temporal evolution. This entails a generic framework for their identification and tracking. The fact that depressions and cyclones are not well-defined objects and their shape and size characteristics change over time makes this task especially challenging. Our method combines the robustness of topological approaches and the detailed tracking information from optical flow analysis. At first cyclones are identified within each time step based on well-established topological concepts. Then candidate tracks are computed from an optical flow field. These tracks are clustered within a moving time window to distill dominant coherent cyclone movements, which are then forwarded to a final tracking step. In contrast to previous methods our method requires only a few intuitive parameters. An integration into an exploratory framework helps in the study of cyclone movement by identifying smooth, representative tracks. Multiple case studies demonstrate the effectiveness of the method in tracking cyclones, both in the northern and southern hemisphere.

EditorsThis January 2018 issue of the IEEE Transactions on Visualization and Computer Graphics contains the proceedings of IEEE VIS 2017, held during 1-6 October 2017. In 2017, IEEE VIS returns to the city of Phoenix, AZ, USA, for the conference's 28th year. The conference will be held at the Hyatt Regency Phoenix hotel. VIS consists of three conferences, held concurrently: the IEEE Visual Analytics Science and Technology Conference (VAST 2017), the IEEE Information Visualization Conference (InfoVis 2017), and the IEEE Scientific Visualization Conference (SciVis 2017). Information on the paper review process is provided along with an overview of each conference.

In this paper, we tackle the challenge of effective Transfer Function (TF) design for Direct Volume Rendering (DVR) of full-color datasets. We propose a novel TF design toolbox based on color similarity which is used to adjust opacity as well as replacing colors. We show that both CIE L*u*v* chromaticity and the chroma component of YCbCr are equally suited as underlying color space for the TF widgets. In order to maximize the area utilized in the TF editor, we renormalize the color space based on the histogram of the dataset. Thereby, colors representing a higher share of the dataset are depicted more prominently, thus providing a higher sensitivity for fine-tuning TF widgets. The applicability of our TF design toolbox is demonstrated by volume ray casting challenging full-color volume data including the visible male cryosection dataset and examples from 3D histology.

We generalize the framework of moments and introduce a definition of invariants for three-dimensional vector fields. To do so, we use the method of moment normalization that has been shown to be useful in the two dimensions. Using invariant moments, we show how to search for patterns in these fields independent from their position, orientation and scale. From the first order vector moment tensor, we construct a complete and independent set of descriptors. We test the invariants in queries on synthetic and real world flow fields.

The analysis of 2D flow data is often guided by the search for characteristic structures with semantic meaning. One way to approach this question is to identify structures of interest by a human observer, with the goal of finding similar structures in the same or other datasets. The major challenges related to this task are to specify the notion of similarity and define respective pattern descriptors. While the descriptors should be invariant to certain transformations, such as rotation and scaling, they should provide a similarity measure with respect to other transformations, such as deformations. In this paper, we propose to use moment invariants as pattern descriptors for flow fields. Moment invariants are one of the most popular techniques for the description of objects in the field of image recognition. They have recently also been applied to identify 2D vector patterns limited to the directional properties of flow fields. Moreover, we discuss which transformations should be considered for the application to flow analysis. In contrast to previous work, we follow the intuitive approach of moment normalization, which results in a complete and independent set of translation, rotation, and scaling invariant flow field descriptors. They also allow to distinguish flow features with different velocity profiles. We apply the moment invariants in a pattern recognition algorithm to a real world dataset and show that the theoretical results can be extended to discrete functions in a robust way.

The analysis of 2D flow data is often guided by the search for characteristic structures with semantic meaning. One way to approach this question is to identify structures of interest by a human observer. The challenge then, is to find similar structures in the same or other datasets on different scales and orientations. In this paper, we propose to use moment invariants as pattern descriptors for flow fields. Moment invariants are one of the most popular techniques for the description of objects in the field of image recognition. They have recently also been applied to identify 2D vector patterns limited to the directional properties of flow fields. In contrast to previous work, we follow the intuitive approach of moment normalization, which results in a complete and independent set of translation, rotation, and scaling invariant flow field descriptors. They also allow to distinguish flow features with different velocity profiles. We apply the moment invariants in a pattern recognition algorithm to a real world dataset and show that the theoretic results can be extended to discrete functions in a robust way.

This paper is the result of a close collaboration of mechanical engineers and visualization researchers. It showcases how interdisciplinary work can lead to new insight and progress in both fields. Our case is concerned with one step in the product development process. Its goal is the design of mechanical parts that are functional, meet required quality measures and can be manufactured with standard production methods. The collaboration started with unspecific goals and first experiments with the available data and visualization methods. During the course of the collaboration many concrete questions arose and in the end a hypothesis was developed which will be discussed and evaluated in this paper. We facilitate a case study to validate our hypothesis. For the case study we consider the design of a reinforcement structure of a brake lever, a plastic ribbing. Three new lever geometries are developed on basis of our hypothesis and are compared against each other and against a reference model. The validation comprises standard numerical and experimental tests. In our case, all new structures outperform the reference geometry. The results are very promising and suggest potential to impact the product development process also for more complex scenarios.

This paper proposes a vector field visualization, which mimics a sketch-like representation. The visualization combines two major perspectives: Large scale trends based on a strongly simplified field as background visualization and a local visualization highlighting strongly expressed features at their exact position. Each component considers the vector field itself and its spatial derivatives. The derivate is an asymmetric tensor field, which allows the deduction of scalar quantities reflecting distinctive field properties like strength of rotation or shear. The basis of the background visualization is a vector and scalar clustering approach. The local features are defined as the extrema of the respective scalar fields. Applying scalar field topology provides a profound mathematical basis for the feature extraction. All design decisions are guided by the goal of generating a simple to read visualization. To demonstrate the effectiveness of our approach, we show results for three different data sets with different complexity and characteristics.

Abstract Tensors provide a powerful language to describe physical phenomena. Consequently, they have a long tradition in physics and appear in various application areas, either as the final result of simulations or as intermediate product. Due to their complexity, tensors are hard to interpret. This motivates the development of well‐conceived visualization methods. As a sub‐branch of scientific visualization, tensor field visualization has been especially pushed forward by diffusion tensor imaging. In this review, we focus on second‐order tensors that are not diffusion tensors. Until now, these tensors, which might be neither positive‐definite nor symmetric, are under‐represented in visualization and existing visualization tools are often not appropriate for these tensors. Hence, we discuss the strengths and limitations of existing methods when dealing with such tensors as well as challenges introduced by them. The goal of this paper is to reveal the importance of the field and to encourage the development of new visualization methods for tensors from various application fields.

We present a new method for the generation of anisotropic sample distributions on planar and two-manifold domains. Most previous work that is concerned with aperiodic point distributions is designed for isotropically shaped samples. Methods focusing on anisotropic sample distributions are rare, and either they are restricted to planar domains, are highly sensitive to the choice of parameters, or they are computationally expensive. In this paper, we present a time-efficient approach for the generation of anisotropic sample distributions that only depends on intuitive design parameters for planar and two-manifold domains. We employ an anisotropic triangulation that serves as basis for the creation of an initial sample distribution as well as for a gravitational-centered relaxation. Furthermore, we present an approach for interactive rendering of anisotropic Voronoi cells as base element for texture generation. It represents a novel and flexible visualization approach to depict metric tensor fields that can be derived from general tensor fields as well as scalar or vector fields.

We propose a combinatorial algorithm to track critical points of 2D time-dependent scalar fields. Existing tracking algorithms such as Feature Flow Fields apply numerical schemes utilizing derivatives of the data, which makes them prone to noise and involve a large number of computational parameters. In contrast, our method is robust against noise since it does not require derivatives, interpolation, and numerical integration. Furthermore, we propose an importance measure that combines the spatial persistence of a critical point with its temporal evolution. This leads to a time-aware feature hierarchy, which allows us to discriminate important from spurious features. Our method requires only a single, easy-to-tune computational parameter and is naturally formulated in an out-of-core fashion, which enables the analysis of large data sets. We apply our method to synthetic data and data sets from computational fluid dynamics and compare it to the stabilized continuous Feature Flow Field tracking algorithm.

Abstract This paper is concerned with the extraction of the surface topology of tensor fields on 2D triangulated manifolds embedded in 3D. In scientific visualization topology is a meaningful instrument to get a hold on the structure of a given dataset. Due to the discontinuity of tensor fields on a piecewise planar domain, standard topology extraction methods result in an incomplete topological skeleton. In particular with regard to the high computational costs of the extraction this is not satisfactory. This paper provides a method for topology extraction of tensor fields that leads to complete results. The core idea is to include the locations of discontinuity into the topological analysis. For this purpose the model of continuous transition bridges is introduced, which allows to capture the entire topology on the discontinuous field. The proposed method is applied to piecewise linear three‐dimensional tensor fields defined on the vertices of the triangulation and for piecewise constant two or three‐dimensional tensor fields given per triangle, e.g. rate of strain tensors of piecewise linear flow fields.

Acceleration is a fundamental quantity of flow fields that captures Galilean invariant properties of particle motion. Considering the magnitude of this field, minima represent characteristic structures of the flow that can be classified as saddle- or vortex-like. We made the interesting observation that vortex-like minima are enclosed by particularly pronounced ridges. This makes it possible to define boundaries of vortex regions in a parameter-free way. Utilizing scalar field topology, a robust algorithm can be designed to extract such boundaries. They can be arbitrarily shaped. An efficient tracking algorithm allows us to display the temporal evolution of vortices. Various vortex models are used to evaluate the method. We apply our method to two-dimensional model systems from computational fluid dynamics and compare the results to those arising from existing definitions.

This paper introduces a novel importance measure for critical points in 2D scalar fields. This measure is based on a combination of the deep structure of the scale space with the well-known concept of homological persistence. We enhance the noise robust persistence measure by implicitly taking the hill-, ridge- and outlier-like spatial extent of maxima and minima into account. This allows for the distinction between different types of extrema based on their persistence at multiple scales. Our importance measure can be computed efficiently in an out-of-core setting. To demonstrate the practical relevance of our method we apply it to a synthetic and a real-world data set and evaluate its performance and scalability.

This paper introduces a novel approximation algorithm for the fundamental graph problem of combinatorial vector field topology (CVT). CVT is a combinatorial approach based on a sound theoretical basis given by Forman's work on a discrete Morse theory for dynamical systems. A computational framework for this mathematical model of vector field topology has been developed recently. The applicability of this framework is however severely limited by the quadratic complexity of its main computational kernel. In this work, we present an approximation algorithm for CVT with a significantly lower complexity. This new algorithm reduces the runtime by several orders of magnitude and maintains the main advantages of CVT over the continuous approach. Due to the simplicity of our algorithm it can be easily parallelized to improve the runtime further.

This work introduces a novel streamline seeding technique based on dual streamlines that are orthogonal to the vector field, instead of tangential. The greedy algorithm presented here produces a net of orthogonal streamlines that is iteratively refined resulting in good domain coverage and a high degree of continuity and uniformity. The algorithm is easy to implement and efficient, and it naturally extends to curved surfaces.

We present a practical approach to generate stochastic anisotropic samples with Poisson-disk characteristic over a two-dimensional domain. In contrast to isotropic samples, we understand anisotropic samples as non-overlapping ellipses whose size and density match a given anisotropic metric. Anisotropic noise samples are useful for many visualization and graphics applications. The spot samples can be used as input for texture generation, e.g., line integral convolution (LIC), but can also be used directly for visualization. The definition of the spot samples using a metric tensor makes them especially suitable for the visualization of tensor fields that can be translated into a metric. Our work combines ideas from sampling theory and mesh generation. To generate these samples with the desired properties we construct a first set of non-overlapping ellipses whose distribution closely matches the underlying metric. This set of samples is used as input for a generalized anisotropic Lloyd relaxation to distribute noise samples more evenly. Instead of computing the Voronoi tessellation explicitly, we introduce a discrete approach which combines the Voronoi cell and centroid computation in one step. Our method supports automatic packing of the elliptical samples, resulting in textures similar to those generated by anisotropic reaction-diffusion methods. We use Fourier analysis tools for quality measurement of uniformly distributed samples. The resulting samples have nice sampling properties, for example, they satisfy a blue noise property where low frequencies in the power spectrum are reduced to a minimum.

Flow vector fields contain a wealth of information that needs to be visualized. As an extension of the well-known streamline technique, we have developed a context-based method for visualizing steady flow vector fields in two and three dimensions. We call our method "Priority Streamlines". In our approach, the density of the streamlines is controlled by a scalar function that can be user-defined, or be given by additional information (e.g., temperature, pressure, vorticity, velocity) considering the underlying flow vector field. In regions, which are interesting the streamlines are drawn with increased density, while less interesting regions are drawn sparsely. Since streamlines in the most important regions are drawn first, we can use thresholding to obtain a streamline representation highlighting essential features. Color-mapping and transparency can be used for visualizing other information hidden in the flow vector field.

We present a novel approach for analyzing two-dimensional (2D) flow field data based on the idea of invariant moments. Moment invariants have traditionally been used in computer vision applications, and we have adapted them for the purpose of interactive exploration of flow field data. The new class of moment invariants we have developed allows us to extract and visualize 2D flow patterns, invariant under translation, scaling, and rotation. With our approach one can study arbitrary flow patterns by searching a given 2D flow data set for any type of pattern as specified by a user. Further, our approach supports the computation of moments at multiple scales, facilitating fast pattern extraction and recognition. This can be done for critical point classification, but also for patterns with greater complexity. This multi-scale moment representation is also valuable for the comparative visualization of flow field data. The specific novel contributions of the work presented are the mathematical derivation of the new class of moment invariants, their analysis regarding critical point features, the efficient computation of a novel feature space representation, and based upon this the development of a fast pattern recognition algorithm for complex flow structures.

Vector field visualization approaches can broadly be categorized into approaches that directly visualize local or integrated flow and approaches that analyze the topological structure and visualize extracted features. Our goal was to come up with a method that falls into the first category, yet reveals structural information. We have developed a dense flow visualization method that shows the overall flow behavior while accentuating structural information without performing a topological analysis. Our method is based on a geometry-based flow integration step and a texture-based visual exploration step. The flow integration step generates a density field, which is written into a texture. The density field is generated by tracing particles under the influence of the underlying vector field. When using a quasi-random seeding strategy for initialization, the resulting density is high in attracting regions and low in repelling regions. Density is measured by the number of particles per region accumulated over time. We generate one density field using forward and one using backward propagation. The density fields are explored using texture-based rendering techniques. We generate the two output images separately and blend the results, which allows us to distinguish between inflow and outflow regions. We obtained dense flow visualizations that display the overall flow behavior, emphasize critical and separating regions, and indicate flow direction in the neighborhood of these regions. We have test our method for isolated first-order singularities and real data sets.

The physical interpretation of mathematical features of tensor fields is highly application-specific. Existing visualization methods for tensor fields only cover a fraction of the broad application areas. We present a visualization method tailored specifically to the class of tensor field exhibiting properties similar to stress and strain tensors, which are commonly encountered in geomechanics. Our technique is a global method that represents the physical meaning of these tensor fields with their central features: regions of compression or expansion. The method is based on two steps: first, we define a positive definite metric, with the same topological structure as the tensor field; second, we visualize the resulting metric. The eigenvector fields are represented using a texture-based approach resembling line integral convolution (LIC) methods. The eigenvalues of the metric are encoded in free parameters of the texture definition. Our method supports an intuitive distinction between positive and negative eigenvalues. We have applied our method to synthetic and some standard data sets, and "real" data from earth science and mechanical engineering application.

To display the intuitive meaning of an abstract metric it is helpful to look on an embedded surface with the same inner geometry as the given metric. The resulting partial differential equations have no standard solution. Only for some special cases satisfactory methods are known. I present a new algorithmic approach which is not based on differential equations. In contrast to other methods this technique also works if the embedding exists only locally. The fundamental idea is to estimate Euclidean distances, from which the surface is built up. In this paper I focus on the reconstruction of a surface from these estimated distances. Particular the influence of a perturbation of the distances on the shape of the resulting surface is investigated.

One of the main research topics in scientific visualization is to "visualize the appropriate features" of a certain structure or data set. Geodesics are very important in geometry and physics, but there is one major problem which prevents scientists from using them as a visualization tool: the differential equations for geodesics are very complicated and in most cases numerical algorithms must be used. There is always a certain approximation error involved. How can you be sure to visualize the features and not only the approximation quality. The paper presents an algorithm to overcome this problem. It consists of two parts. In the first, a geometric method for the construction of geodesics of arbitrary surfaces is introduced. This method is based on the fundamental property that geodesics are a generalization of straight lines on plains. In the second part these geodesics are used to generate local nets on the surfaces.